Interesting zeros of
Many sequences pn(x) of polynomials seem to
have interesting sets of (complex) zeros. Here are some examples.
- Let Kn(x) denote the sum over all
connected simple graphs on an n-element vertex set of
xe(G)-n+1, where e(G) denotes the number
of edges of G. For instance, K3(x) = 3 +
x. Zeros of K14(x): ps or pdf.
- Let Xm,n(q) denote the chromatic polynomial
of the complete bipartite graph
Km,n. Equivalently, Xm,n(q)
is the coefficient of xmyn/m!n! in the
power series expansion about x=y=0 of
(ex + ey -1)q. Zeros of
X20,20(q): ps or pdf. Note.
Appearances to the contrary, the integers 2, 3, 4, 5, 6, 7, 8, 9 are
not zeros of X20,20(q). For instance, the zero
close to q=2 is actually
- Let Cn(q) denote the sum of
qA(P)over all lattice paths P from (0,0)
to (n,n) with steps (0,1) and (1,0), such that P
never rises above the line y=x, and where A(P) is
the area under the path (and above the x-axis). The number
of such paths is the Catalan number Cn, and
Cn(q) is a standard q-analogue of
Cn (see Enumerative Combinatorics,
vol. 2, Exercise 6.34(a)). For instance, C3(q) = 1 + q
+ 2q2 + q3. Zeros of
C32(q): ps or pdf.
- Let Hn(r) denote the number of
matrices of nonnegative integers such that every row and column sums
to r. For instance, Hn(0)=1,
Hn(1)=n!, and H2(r)=r+1. It
is known that for fixed n, Hn(r) is a
polynomial in r of degree (n-1)2. (See
for instance Enumerative Combinatorics, vol. 1, Proposition
4.6.19.) Zeros of H9(r): ps or pdf.
- (suggested by G. Labelle) Let
Nn(x) be the sum of
(1/n)φ(d)xn/d over all (positive) divisors
d of n, where φ(d) denote Euler's
phi-function. If x is a positive integer, then
Nn(x) is the number of "necklaces" of length
n with x colors, i.e., the number of sequences, up
to cyclic shift, with n terms from an alphabet of
x letters. Zeros of N120(x): ps or pdf. Similar
behavior is seen when φ is replaced by μ (the Möbius function
of number theory), in which case necklaces are replaced by
primitive necklaces (those with no symmetry).
- (suggested by H. Wilf) Let
Bn(x) denote the nth Bernoulli polynomial.
Zeros of B300(x): pdf (corrected on 5/2/04). Click here (pdf file) for information on the real zeros of
Bn(x). For the limiting curve formed by the
complex zeros, click here.
- Let pk(n) denote the number of partitions of
the positive integer n into k parts. Let
Fk(x) denote the sum over all n of
pk(n)xn. For instance,
F4(x) = x4 +
x3 +2x2+ x.
Zeros of F200(x): ps or pdf.
- Zeros of the sum from 1 to 300 of